# Integral of e^(x^2 + y^2) dx dy over x from 0 to sqrt(25 - y^2) and over y from -5 to 5. (a)...

## Question:

{eq}\displaystyle \int_{-5}^{5} \int_{0}^{\sqrt{25 - y^2}} e^{(x^2 + y^2)}dx \, dy {/eq}

(a) Sketch the region of integration.

(b) Change to polar coordinates.

(c) Evaluate the integral.

## Polar Coordinates:

When faced with a double integral, recall that not only can we switch the order of integration but we can switch coordinate systems entirely. For the situation above, it will be more convenient to use polar coordinates.

Recall

{eq}x = r \cos \theta {/eq}

{eq}y = r \sin \theta {/eq}

{eq}r^2 = x^2+y^2 {/eq}

{eq}\theta = \tan^{-1} \frac{y}{x} {/eq}

{eq}dA = r\ dr\ d\theta {/eq}

## Answer and Explanation:

**Part A**

From the limits on the integral, we see that the region of integration is {eq}0 \leq x \leq \sqrt{25-y^2} {/eq} when {eq}y \in [-5,5] {/eq}....

See full answer below.

Become a Study.com member to unlock this answer! Create your account

View this answer#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

from Precalculus: High School

Chapter 24 / Lesson 1